Ome new theorems on generating functions and their applications on odd and even certain numbers attached to p and q parameters

TL;DR

This paper introduces new theorems using symmetrizing operators to derive novel generating functions for various (p,q)-number sequences, including Fibonacci, Lucas, Pell, and Jacobsthal numbers, and their products with odd and even terms.

Contribution

It presents new theorems and generating functions for (p,q)-number sequences, expanding the mathematical tools available for these generalized sequences.

Findings

New theorems using symmetrizing operators

Generating functions for (p,q)-number sequences

Generating functions for products with odd and even terms

Abstract

In this study, we first provide some new theorems by using the symmetrizing operator. After that, by using this theorems we introduce a new family of generating functions of odd and even terms of (p,q)-Fibonacci numbers, (p,q)-Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Then, we give the new generating functions of the products of these (p,q)-numbers with odd and even phrases of (p,q)-numbers.